More

1. "Conditional Quantum Dynamics and Logic Gates from Oxford's David Deutsch et-al, asserts that "the computational power of quantum computers exceeds that of Turing machines". The word "conditional" means that "one subsystem undergoes a coherent evolution that depends on the quantum state of another subsystem". The two subsystems are called "control" (i.e, "C") and "target" (i.e.,"T"). The quantum bits might be the single electrons controlling the spatial configurations of the dimers on microtubules, for example. What about spacelike influences of the EPR type between the two?

The paper focuses on a "quantum controlled-NOT gate" labeled as "C(control,target)". The classical limit of this NOT gate is reversible acting on the target bit and the control bit. In the classical limit, neither the control nor the target bit are in a coherent superposition of some preferred "pointer basis" defined by distant correlations with the environment. The value of the target bit is negated if the control bit is "on", otherwise the target bit is left unchanged.

C(control,target) is allegedly a unitary operator at the quantum level. Given a common basis {|0>,|1>} for each of the two quantum bits (control and target),

C(control,target)|control>|target> = |control>|control+target mod2>

The symbol "|control>|control+target mod2>" means that the final eigenvalue of the target quantum bit is the sum of the eigenvalue of the control bit with the initial eigenvalue of the target bit. These eigenvalues are "0" and "1". So the "truth table" is: For control = 0: target 0 -> 0, target 1 -> 1; for control = 1, target 0 -> 1, target 1 -> 0. The new feature is the quantum superposition principle. For example, let the initial control (target) bits be the coherent superpositions

|coni> = |0><0|coni> + |1><1|coni>

|targi> = |0><0|targi> + |1><1|targi>

Therefore:

C(control,target)|coni>|targi> = C..[|0>|0>..+ |1>|1>..+ |1>|0>..+ |0>|1>..]

= |0>|0>.. + |1>|0>.. + |1>|1>..+ |0>|1>..

= |0>|0><0|coni><0|targi> + |1>|0><1|coni><1|targi>

+ |1>|1><1|coni><0|targi> + |0>|1><0|coni><1|targi>

This is generally an entangled Einstein-Podolsky-Rosen (EPR) pair state with nonlocal spacelike influences between the control and the target subystems in violation of Bell's locality inequality also characteristic of a "measurement gate". That is, the quantum controlled-NOT logic gate converts an uncorrelated pair of quantum bits into a correlated pair which is also what happens in quantum measurement theory. Let's work out some examples:

That is, initially the target is classically "off", but the control is in a quantum coherent superposition of both "on" and "off". Therefore:

C..|coni>|targi> = |0>|0><0|coni> + |1>|1><1|coni>

This is an EPR entangled state. In the final state either both control and target are each "on" or each "off" with the same probabilities that the control had for being "on" or "off". The control bit has forced the target bit into a kind of possibly distant clone of itself. This is reminiscent of excitatory association in neural nets. It can do this faster-than-light (FTL) and even backward-in-time (BIT) depending on the frame-invariant spacetime interval between local irreversible measurements of the eigenvalue of each quantum bit.

That is, initially the target is classically "on", but the control is in a coherent quantum superposition as in example 1.

C..|coni>|targi> = |1>|0><1|coni> + |0>|1><0|coni>

This is also an EPR entangled state. Now the control has made a kind of inhibitory anti-clone of itself. When the control is "on" the target is "off" and vice-versa.

That is, the control is classically "off" and the target is in a generic quantum superposition. This is the trivial identity transformation for the target bit. Is there a group structure here? That is, can we define group elements for the independent two-fold continuous complex parameters <0|coni> and <0|targi>?

That is, the control is classically "on" and the target is in a generic quantum superposition.

C.. |coni>|targi> = |1>|0><1|targi> + |1>|1><0|targi> = |1>[|0><1|targi> + |1><0|targi>] which is a kind of "spin flip" of the target quantum bit. The probabilities are exchanged in the target Hilbert subspace, and there is no EPR entanglement between control and target subsystems. Nevertheless, the action-at-a-distance (AAAD) of the control on the target can be locally detected if <1|targi> is not equal in magnitude to <0|targi>.

We should further investigate what happens if we choose to measure in other bases like

|yes> = (1/rt2){|on> + |off>} and |no> = (1/rt2){|on> - |off>} for either the target or the control or both.

The authors claim:

This transformation of superpositions into entanglements can be reversed by applying the same controlled-NOT operation again. Hence it can be used to implement the so-called Bell measurement on the two bits by disentangling the Bell states. [Phys. Rev. Lett 68, 3259, 1992].

The authors assert:

The realization of the Bell measurement on the two qubits is the main obstacle to the practical implementation of quantum teleportation and dense quantum coding.

Quantum state swapping can be achieved by cascading three quantum controlled-NOT gates.

The quantum controlled-NOT gate may also be used to swap distantly separated states in the presence of a channel carrying only classical information.

The remarkable feature of all these processes is that in the presence of shared entanglement an arbitrary state |x> may be transferred as a result of sending only a few bits of classical information despite the fact that |x> depends on two continuous parameters corresponding to an infinite amount of classical information. p.4084

The quantum NOT gate is not universal, but a universal quantum gate is constructable. (Their ref 11).

The authors suggest two physical mechanisms to implement their quantum gate. They are Ramsey atomic interferometry, and the selective driving of optical resonances using the dipole-dipole interaction.

Tycho Sleator and Harald Weinfurter of NYU wrote "Realizable Universal Quantum Logic Gates" in Phys Rev Letters 74, p.4087 (1995). They give us a universal 2-bit quantum gate. Cavity quantum electrodynamics may be used to make this gate. These gates require single particle resolution.

Are the single electrons controlling the conformation of each protein dimer in the microtubules of living cells such universal quantum gates forming the biocomputing physical substrate of our minds?

A quantum Turing machine is described in Proc Roy Soc, A400, 96 (1985). Factoring takes exponential time on a classical computer but only polynomial time on a quantum computer for a given algorithm. The state of a classical computer can be known during the computation. This is thought not to be possible in a quantum computer because a von Neumann "collapse" type state determination will generally disrupt the coherence. Is a new kind of non-demolition or a protected measurement possible?

The universal quantum gate (which can simulate any non-Boolean quantum logic gate), like its classical Boolean offspring, is reversible. Reversibility here is identified with the unitarity of the time evolution matrix. Classical universal gates (Fredkin, Toffoli) need three input bits and three output bits. An optical implementation is in Phys Rev Letters, 62, 2124 (1989). An n-bit quantum gate is a matrix Sq transforming state vectors in 2^n dimensional Hilbert space.

A 3-bit universal quantum gate called the Deutsch gate is represented by an 8x8 matrix (e.g., eq 2, p. 4087). A 2-bit gate by a 4x4 matrix. The universal 2-bit gate is a generalized "measurement gate" as described in the first paper above by Deutsch et-al. As described above, the measurement gate C(control,target) is defined by invariance of the control bit eigenvalue while the target bit out eigenvalue is the Boolean sum of the input control bit eigenvalue with the input target bit eigenvalue. The quantum measurement gate is the matrix Sm

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

i.e., eq. 3 p.4088

The universal quantum gate is the matrix Su(t)

1 0 0 0

0 1 0 0

0 0 e^ipi/4 cos(pit/2) e^-ipi/4 sin(pit/2)

0 0 e^-ipi/4 sin(pit/2) e^ipi/4 cos(pit/2)

i.e., eq. 4

If the arbitrary "rotation" parameter t = 1, the universal gate is the square root of the measurement gate. The 3-bit Deutsch gate is decomposed into a network of universal 2-bit gates and measurement gates (e.g., eq 5 and Fig. 1 p.4088). Measurement gates correspond to a kind of nondemolition (i.e., QND) measurement according to the authors of both papers considered here so far.

Cavity quantum electrodynamics is to our knowledge the only candidate capable of realizing quantum logic gates in the near future... QND measurements can achieve single particle (photon) resolution, a condition necessary for the construction of a universal quantum gate.

From: qix@desire.apana.org.au

To: omega-point-theory@world.std.com, aleph-l@netcom.com

Subject: Q-mind theories and strings Reply-To: omega-point-theory@world.std.com

I have just run across the following paper at http://xxx.lanl.gov/abs/hep-ph/9505374 :

Title: Theory of Brain Function, Quantum Mechanics and Superstrings

Author: D. Nanopoulos

Comments: 72 pages, 1 figure (uuencoded)

Report-no: CERN-TH/95-128

E-mail: dimitri@phys.tamu.edu or nanopoud@cernvm.cern.ch

Unfortunately the Abstract at that URL has been truncated, so I shall simply quote from the start of the paper.

--- begin quote ---

Abstract

Recent developments/efforts to understand aspects of the brain function at the sub-neural level are discussed. MicroTubules (MTs), protein polymers constructing the cytoskeleton, participate in a wide variety of dynamical processes in the cell. Of special interest to us is the MTs participation in bioinformation processes such as learning and memory, by possessing a well-known binary error-correcting code [$K_1(13,2^6,5)$] with 64 words. In fact, MTs and DNA/RNA are unique cell structures that possess a code system. It seems that the MTs' code system is strongly related to a kind of ``Mental Code" in the following sense. The MTs' periodic paracrystalline structure make them able to support a superposition of coherent quantum states, as it has been recently conjectured by Hameroff and Penrose, representing an external or mental order, for sufficient time needed for efficient quantum computing. Then the quantum superposition collapses spontaneously/dynamically through a new, string-derived mechanism for collapse proposed recently by Ellis, Mavromatos, and myself.

At the moment of collapse, organized quantum exocytosis occurs, ie, the simultaneous emission of neurotransmitter molecules by the synaptic vesicles, embedded in the ``firing zone" of the presynaptic vesicular grids. Since in the superposition of the quantum states only those participate that are related to the ``initial signal", when collapse occurs, it only enhances the probability for ``firing" of the relevant neurotransmitter molecules. That is how a ``mental order" may be translated into a ``emphysiological action". Our equation for quantum collapse, tailored to the MT system, predicts that it takes 10,000 neurons ${\cal O}(1\,{\rm sec})$ to dynamically collapse, in other words to process and imprint information.

[Comment by Jack Sarfatti. Stu Hameroff tells me that Penrose also gets 10,000 neurons as the minimum number for a conscious event via his graviton collapse criterion.]

Different observations/experiments and various schools of thought are in agreement with the above numbers concerning ``conscious events". If indeed MTs, with their fine structure, vulnerable to our quantum collapse mechanism may be considered as the microsites of consciousness, then several, unexplained (at least to my knowledge) by traditional neuroscience, properties of consciousness/awareness, get easily explained, including ``backward masking", `` referal backwards in time", etc. Furthermore, it is amusing to notice that the famous puzzle of why the left (right) part of the brain coordinates the right (left) part of the body, ie, the signals travel maximal distance, is easily explained in our picture. In order to have timely quantum collapse we need to excite as much relevant material as possible, thus signals have to travel the maximal possible distance. The non-locality in the cerebral cortex of neurons related to particular missions, and the related unitary sense of self as well as non-deterministic free will are consequences of the basic principles of quantum mechanics, in sharp contrast to the ``sticks and balls" classical approach of conventional neural networks. The proposed approach clearly belongs to the reductionist school since quantum physics is an integrated part of our physical world. It is highly amazing that string black-hole dynamics that have led us to contemplate some modifications of standard quantum mechanics, such that the quantum collapse becomes a detailed dynamical mechanism instead of being an ``external" ad-hoc process, may find some application to some quantum aspects of brain function. It looks like a big universality principle is at work here, because both in the black hole and the brain we are struggling with the way information is processed, imprinted, and retrieved.

--- end quote ---

The string-inspired collapse mechanism to which Nanopoulos refers is a generalization of the density-matrix formalism of quantum mechanics, in which (as far as I can see) pure quantum states evolve stochastically (in the conventional theory they evolve deterministically).

-mitch

http://desire.apana.org.au/~qix

Return to Star Fleet